Subgroups — #26109124

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688873	C₂	1	2	yes	maximal, complemented
4	#20241222	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
6	#15689176	—	2	4	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	#15688878	—	2	1	no	—
14	#15688873	—	2	4	no	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₄₇×#336	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₄⋊C₈×C₄₇	—	2	2	no	—
41	C₂²⋊C₈×C₄₇	—	4	2	no	—
42	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	1	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	4	1	no	—
51	C₂²×C₄×C₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	—	4	2	no	—
63	C₄²×C₄₇	—	2	1	no	—
64	C₄²×C₄₇	—	2	1	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	C₂×D₄	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₂×C₄₇⋊C₈	—	2	2	no	—
70	C₂×C₄₇⋊C₈	—	2	2	no	—
71	C₂×C₄₇⋊C₈	—	2	2	no	—
72	C₂×C₄₇⋊C₈	—	2	2	no	—
73	C₂×C₄₇⋊C₈	—	2	2	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
78	C₂³×C₄₇	—	2	1	no	—
79	C₂³×C₄₇	—	2	1	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	2	1	no	—
83	C₂³×C₄₇	—	2	1	no	—
84	C₂³×C₄₇	#106	1	2	yes	—
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	1	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	#106	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#121	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	#131	1	2	yes	—
107	C₂×C₄×C₄₇	#131	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#135	1	2	yes	—
111	C₈×C₄₇	—	8	2	no	—
112	C₄₇⋊C₈	—	2	4	no	—
113	C₄₇⋊C₈	—	4	2	no	—
114	C₄₇⋊C₈	—	8	2	no	—
115	#8598	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	4	1	no	—
120	C₂²×C₄₇	—	4	1	no	—
121	C₂²×C₄₇	—	4	1	no	—
122	C₂²×C₄₇	—	4	1	no	—
123	C₂²×C₄₇	#380	1	1	yes	characteristic
124	C₂²×C₄₇	#424	1	1	yes	characteristic
125	C₂²×C₄₇	#424	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
127	C₂²×C₄₇	#427	1	1	yes	characteristic
128	C₂²×C₄₇	#430	1	1	yes	characteristic
129	C₂²×C₄₇	#430	1	1	yes	characteristic
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	4	1	no	—
133	C₄×C₄₇	—	4	1	no	—
134	C₄×C₄₇	#408	1	2	yes	—
135	C₄×C₄₇	#433	1	2	yes	—
136	C₄×C₄₇	#465	1	2	yes	—
137	C₄×C₄₇	#465	1	2	yes	—
138	C₂×#431	—	47	1	no	—
139	#1797	—	47	1	no	—
140	#1797	D₄₇	1	1	yes	characteristic
141	C₂×#334	—	47	2	no	—
142	#1277	—	47	2	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#1570	1	1	yes	characteristic
145	C₂×C₄₇	#1570	1	1	yes	characteristic
146	C₂×C₄₇	#1572	1	1	yes	characteristic
147	C₂×C₄₇	#1572	1	1	yes	characteristic
148	C₂×C₄₇	#1798	1	1	yes	characteristic
149	C₂×C₄₇	#1807	1	1	yes	characteristic
150	C₂×C₄₇	#1811	1	1	yes	characteristic
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	#431	—	94	4	no	—
153	C₂×C₄⋊C₈	—	47	2	no	—
154	C₂×C₄⋊C₈	—	47	2	no	—
155	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
156	C₂×C₂²⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	47	2	no	—
158	C₂×C₂²⋊C₈	—	94	1	no	—
159	C₂×C₄×C₈	—	47	1	no	—
160	C₂×C₄×C₈	—	47	1	no	—
161	#336	—	2	1	no	—
162	#336	—	94	1	no	—
163	#334	—	94	4	no	—
164	C₄₇	#8598	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
166	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
167	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	94	4	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₄⋊C₈	—	94	2	no	—
183	C₂²⋊C₈	—	4	2	no	—
184	C₂²⋊C₈	—	94	4	no	—
185	C₂²⋊C₈	—	188	2	no	—
186	C₄×C₈	—	47	4	no	—
187	C₄×C₈	—	94	2	no	—
188	C₂⁴	Dic₄₇⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	#10625335	1	2	yes	—
202	C₂²×C₄	#10625336	1	2	yes	—
203	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	47	4	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	94	2	no	—
214	C₂×C₈	—	94	2	no	—
215	C₂×C₈	—	188	2	no	—
216	C₂×C₈	—	188	2	no	—
217	C₂×C₈	—	188	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	D₄×D₄₇	1	2	yes	—
223	C₄²	#10625352	1	2	yes	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	#13093955	1	2	yes	—
231	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	#13093955	1	2	yes	—
248	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₄	#13093980	1	1	yes	characteristic
250	C₂×C₄	#13093996	1	2	yes	—
251	C₂×C₄	#13094001	1	2	yes	—
252	C₂×C₄	#13094030	1	2	yes	—
253	C₂×C₄	#13094032	1	2	yes	—
254	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
255	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
256	C₂×C₄	#13094044	1	2	yes	—
257	C₈	—	8	2	no	—
258	C₈	—	94	4	no	—
259	C₈	—	376	2	no	—
260	C₈	—	188	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#15688981	1	1	yes	characteristic
269	C₂²	#15689149	1	1	yes	characteristic
270	C₂²	#15689151	1	1	yes	characteristic
271	C₂²	C₂×#13093955	1	1	yes	characteristic
272	C₂²	#15689159	1	1	yes	characteristic
273	C₂²	#15689171	1	1	yes	characteristic
274	C₂²	#15689171	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	#15689088	1	2	yes	—
280	C₄	#15689181	1	2	yes	—
281	C₄	#15689350	1	2	yes	—
282	C₄	#15689354	1	2	yes	—
283	C₂	—	4	2	no	—
284	C₂	#20242158	1	1	yes	characteristic
285	C₂	#20242160	1	1	yes	characteristic
286	C₂	#20242166	1	1	yes	characteristic
287	C₂	#20242169	1	1	yes	characteristic
288	C₂	#20243370	1	1	yes	characteristic
289	C₂	#20243411	1	1	yes	characteristic
290	C₂	#20243428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#8598
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#15688873	C₂	semidirect
4. #20241222	C₂	semidirect
164. C₄₇	#8598	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689176	2
#20243368	2
C₂×#15688873	2
#20241222	2
C₄₇×#1797	2
#8598	47